Find a particular solution to y4y3y13te4tSolutionForm of par

Find a particular solution to y??+4y?+3y=13te^(4t)

Solution

Form of particular solution is

yp=(At+B)e^{4t} based on inhomogeneous part 13te^{4t}

yp\'=(4At+A+4B)e^{4t}

yp\'\'=(16At+8A+16B)e^{4t}

SUbstituting gives

16At+8A+16B+16At+4A+16B+3At+3B=13t

Comparing coefficinets gives

35A=13 hence, A=13/35

12A+35B=0

B=-12A/35=-156/35^2

yp=(13t/35-156/35^2)e^{4t}